A system is characterized by the following state equation and output equation ( $u$ : input,

$x$ : state vector, $y$ : output)

$$ \begin{aligned} & \dot{x}=\left[\begin{array}{cc} a & b \\ -a & 0 \end{array}\right] x+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] u \\ & y=\left[\begin{array}{ll} 1 & 2 \end{array}\right] x \end{aligned} $$

What are the values of $a$ and $b$ for which the poles of the transfer function are at $-2+j 3$ and $-2-\beta$ ?

A system is represented in state-space form as follows:

(u: input, $x$ : state vector, $y$ : output)

$$ \begin{aligned} & \dot{x}=\left[\begin{array}{cc} 1 & 2 \\ -3 & 0 \end{array}\right] x+\left[\begin{array}{l} 1 \\ 2 \end{array}\right] u \\ & y=\left[\begin{array}{ll} 1 & 2 \end{array}\right] x \end{aligned} $$

Consider the new state vector $z=\left[\begin{array}{cc}2 & 1 \\ -1 & 0\end{array}\right] x$

What is the state-space representation of the system in terms of the new state vector $z$ ?

Consider the state-space model $$ \begin{aligned} \dot{x}(t) & =A x(t)+B u(t) \\ y(t) & =C x(t) \end{aligned} $$ where $x(t), r(t), y(t)$ are the state, input and output, respectively. The matrices $A, B, C$ are given below $$ A=\left[\begin{array}{cc} 0 & 1 \\ -2 & -3 \end{array}\right], B=\left[\begin{array}{l} 0 \\ 1 \end{array}\right], C=\left[\begin{array}{ll} 1 & 0 \end{array}\right] $$ The sum of the magnitudes of the poles is ___________. (Round off to nearest integer)

Consider the state-space description of an LTI system with matrices

$$A = \left[ {\matrix{ 0 & 1 \cr { - 1} & { - 2} \cr } } \right],B = \left[ {\matrix{ 0 \cr 1 \cr } } \right],C = \left[ {\matrix{ 3 & { - 2} \cr } } \right],D = 1$$

For the input, $$\sin (\omega t),\omega > 0$$, the value of $$\omega$$ for which the steady-state output of the system will be zero, is ___________ (Round off to the nearest integer).